Nonlinear Fourier classification of 663 rogue waves measured in the Philippine Sea

Rogue waves are sudden and extreme occurrences, with heights that exceed twice the significant wave height of their neighboring waves. The formation of rogue waves has been attributed to several possible mechanisms such as linear superposition of random waves, dispersive focusing, and modulational instability. Recently, nonlinear Fourier transforms (NFTs), which generalize the usual Fourier transform, have been leveraged to analyze oceanic rogue waves. Next to the usual linear Fourier modes, NFTs can additionally uncover nonlinear Fourier modes in time series that are usually hidden. However, so far only individual oceanic rogue waves have been analyzed using NFTs in the literature. Moreover, the completely different types of nonlinear Fourier modes have been observed in these studies. Exploiting twelve years of field measurement data from an ocean buoy, we apply the nonlinear Fourier transform (NFT) for the nonlinear Schrödinger equation (NLSE) (referred to NLSE-NFT) to a large dataset of measured rogue waves. While the NLSE-NFT has been used to analyze rogue waves before, this is the first time that it is systematically applied to a large real-world dataset of deep-water rogue waves. We categorize the measured rogue waves into four types based on the characteristics of the largest nonlinear mode: stable, small breather, large breather and (envelope) soliton. We find that all types can occur at a single site, and investigate which conditions are dominated by a single type at the measurement site. The one and two-dimensional Benjamin-Feir indices (BFIs) are employed to examine the four types of nonlinear spectra. Furthermore, we verify on a part of the data set that for the localized types, the largest nonlinear Fourier mode can be attributed directly to the rogue wave, and investigate the relation between the height of the rogue waves and that of the dominant nonlinear Fourier mode. While the dominant nonlinear Fourier mode in general only contributes a small fraction of the rogue wave, we find that soliton modes can contribute up to half of the rogue wave. Since the NLSE does not account for directional spreading, the classification is repeated for the first quartile with the lowest directional spreading for each type. Similar results are obtained.

Reply: Thank you for the The water depth there is actually around 5000 m.We have corrected the text.4. Line 280: repetition "contain these cases as special cases".
Reply: We have revised it to "Recently, large families of breathers that contain these solutions as special cases were investigated in the Refs.[105,106]". 5. Line 313: references to panes (k) and (l) are wrong; should be (h) and (i).Please check.
Reply: Thank you for noticing these typos.We have fixed them.
Reply: Thanks for your comment.We have fixed this mistake.

7.
According to the description in the text, the main spectrum is represented by complex numbers Ek (Eq.( 2)) plotted in Fig. 4 with blue circles.No explanation is given how the spines appear in Fig. 4.
Reply: Thank you for bringing this up.To clarify the origin and role of the spines, we extended the final paragraph of the subsection "Operation of periodic NLSE-NFT" as follows.
"The main spectrum consists of the E k , which are connected by curves known as spines.Just like the main spectrum, the spines remain constant during propagation with respect to the NLSE.In principle, spines are redundant because the signal is already uniquely specified by the main and auxiliary spectrum together with the sheet indices.However, they play an important role when the Riemann theta form of the solution in (6) is computed.Spines provide valuable information about the general characteristics of the solution u(X, T ), as they impose topological constraints on the trajectories of the auxiliary spectra µ k (X, T ) [2, p. 49].Like the main and auxiliary spectrum, the spines can be found by spectral analysis of the Zakharov-Shabat operator."8. Line 370.Frankly speaking, I cannot understand how the statement "The difference between the spectral components of sine waves and Stokes waves is that the main spectrum points of sine waves are connected by vertical spines, and the main spectrum points of Stokes waves are connected by distorted spines" may be true.As I may understand, in the present context the difference between sinusoidal waves and Stokes waves is in the presence of phase-locked (bound wave) components which modify the wave shape.The NLS equation describes only the spectral band of the carrier, thus does not contain information about bound Therefore, the distorted spines should reflect something different.I guess, distorted spines may show a significant nonlinear correction to the frequency (?), what is an implicit sign of a Stokes wave.
Reply: Thank you for these comments.The Stokes waves turn into sine waves as their amplitudes become smaller and thus, indeed, nonlinear effects decrease.At the same time, the spines straighten.We are not aware of a mathematical derivation of this phenomenon, but the effect has been consistently observed in the literature.We have added the following clarification to the paper."(The Stokes waves turn into sine waves for small amplitudes.The spines become vertical lines in that case.See e.g.[3,Fig. 13] or compare Figs. 3 and 4 in [1].)" 9. Figure 9 is first mentioned before Fig. 8 (lines 465 and 470 respectively).
Reply: Thank you for noticing this.We have changed the order of two paragraphs.
10. Line 626.The conclusion "Our results therefore do not suggest a connection between the BFI's and the rogue wave probability" is probably not accurate.Such a statement could be drawn based on the scatter AI vs BFI, which is not present in the paper.Based on Fig. 11, one may say that the maximum wave height grows when BFI increases, thus this index may be useful.
Reply: Thank you.To make the argument clearer, we have revised that part to "If the BFIs were correlated with the rogue wave probability, one would expect higher BFIs for the rogue wave data.Since that does not appear to be the case here, our results do not suggest a connection between the BFIs and the rogue wave probability."